2
$\begingroup$

What are some interesting examples of accessible $\infty$-categories which are not presentable and not small?

By interesting I mean a category which comes up naturally in a certain context and in a way which doesn't allow working around it.

I prefer examples from derived algebraic geometry as these will most likely be more understandable to me but any interesting example will do.

Edit: As Dimitri pointed out in a comment there are already examples in ordinary categories (e.g. the category of fields). To focus my question I'd like to add the restriction that the category should not be $1$-truncated.

Improve this question
$\endgroup$
16
  • 4
    $\begingroup$ The category of fields. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 4, 2017 at 3:27
  • 1
    $\begingroup$ Flat modules over a commutative ring spectrum? $\endgroup$
    – Charles Rezk
    Commented Feb 4, 2017 at 18:56
  • 3
    $\begingroup$ @SaalHardali sure. Actually, I'm pretty sure you can embed any accessible $\infty$-category fully inside a presentable one. $\endgroup$
    – Charles Rezk
    Commented Feb 4, 2017 at 19:09
  • 1
    $\begingroup$ I think we shouldn't have this discussion in the comments (it is getting too long), but I do not understand what kind of application you have in mind. For example, as Charles Rezk already told you, every accessible ∞-category embeds into a presentable ∞-category via an accessible functor. $\endgroup$
    – Denis Nardin
    Commented Feb 4, 2017 at 19:38
  • 1
    $\begingroup$ I would very much like to know in what sense "pretty much any type of $\infty$-category can be embedded into a presentable $\infty$-category." I find this statement very hard to believe. The way I see it, most model categories are not combinatorial, and this is a restrictive hypothesis. When I try to read $\infty$-categorical work, it often comes down to assuming things are presentable, which seems an equivalently restrictive hypothesis. This is my main concern about working with $\infty$-categories. $\endgroup$
    – David White
    Commented Feb 4, 2017 at 21:11

0

Reset to default

You must log in to answer this question.